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μ
·
μ
+
μ
·
μ
=
μ
0
,
[
]
there are no triplets
(
T
,
S
,
N
)
for which it can hold
S
(
T
(
a
,
a
),
T
(
a
,
N
(
a
)))
=
0for
all
a
in
.
In the same vein, there are some laws that have solutions when different t-norms,
t-conorms and strong negations are considered. For example,
[
0
,
1
]
(μ
+
μ)
·
(μ
·
μ
)
=
μ
0
,
A
c
that comes from
(
A
∪
A
)
∩
(
A
∩
)
= ∅
, translated in the form
T
1
(
S
(
a
,
a
),
T
2
(
0, has infinite solutions like, for example, with an strong nega-
tion
N
, such that
N
a
,
N
(
a
)))
=
N
0
,
T
1
=
min
,
T
2
=
W
and any t-conorm
S
, since
min
(
S
(
a
,
a
),
T
2
(
a
,
N
(
a
))
=
T
2
(
a
,
N
(
a
))
=
W
(
a
,
N
(
a
))
=
max
(
0
,
a
+
N
(
a
)
−
1
)
=
0
,
because of
T
2
(
a
,
N
(
a
))
a
S
(
a
,
a
)
, and
N
(
a
)
1
−
a
,or
a
+
0.
Another case is given by the classical (derived) laws
N
(
a
)
−
1
A
c
A
c
A
∩
(
∪
B
)
=
A
∩
B
,
A
∪
(
∩
B
)
=
A
∪
B
,
and the corresponding 'possible' fuzzy laws
μ
·
(μ
+
˃)
=
μ
·
˃, μ
+
(μ
·
˃)
=
μ
+
˃,
which functional equations
T
1
(
,
(
(
),
))
=
T
2
(
,
),
S
1
(
,
(
(
),
))
=
S
2
(
,
),
a
S
N
a
b
a
b
a
T
N
a
b
a
b
do not have solutions with
T
1
=
T
2
and
S
1
=
S
2
, respectively, but that with
N
=
N
0
,
W
and
W
∗
do verify
W
∗
(
•
W
(
a
,
1
−
a
,
b
))
=
max
(
0
,
min
(
a
,
b
))
=
min
(
a
,
b
)
W
∗
(
•
a
,
W
(
1
−
a
,
b
))
=
min
(
1
,
max
(
a
,
b
))
=
max
(
a
,
b
)
W
∗
,
W
∗
,
that is, they have the solutions
(
T
1
=
W
,
S
=
T
2
=
min
)
and
(
S
1
=
T
=
W
, respectively. Thus, it is possible to consider more complex algebras
of fuzzy sets by means of n-tuples of the type
,
S
2
=
max
)
(
T
1
,...,
T
m
;
S
1
,...,
S
r
;
N
1
,...,
N
p
)
.
that have no solutions
neither in standard algebras, nor with different t-norms, t-conorms, or different strong
negations. The fact that no standard algebra of fuzzy sets is a Boolean algebra, makes
impossible to simultaneously deal in such algebras with all formulas that are valid
with classical sets.
Notwithstanding, there are more derived laws than
[
]
2.2.9 Examples
Example 2.2.47
In a scale between 10 and 50
ⓦ
C, the label 'cold' referred to
temperature, is graduated by
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